**4. Optimal Convergence Control Parameters**

The parameters *hf* and *hθ* are called convergence control parameters that are computed using the numerical BVPh2.0 package. Resulting optimal numerical values of these parameters are usually determined by the min of the average error. To significantly reduce the processing time of CPU, the tactic of average residual error is used at the m*th*-order of approximation, such that,

$$
\varepsilon\_m^f \left( \hbar\_f \right) = \frac{1}{N+1} \sum\_{j=0}^N \left[ \sum\_{i=0}^k (f\_i)\_{\eta=j\pi} \right]^2 \tag{25}
$$

$$\varepsilon\_{m}^{\theta} \left( \hbar\_{f}, \hbar\_{\theta} \right) = \frac{1}{N+1} \sum\_{j=0}^{N} \left[ \sum\_{i=0}^{k} (f\_{i})\_{\eta = j\pi'} \sum\_{i=0}^{k} (\theta\_{i})\_{\eta = j\pi} \right]^{2} \tag{26}$$

and

$$
\varepsilon\_m^{\theta} \left( \hbar\_{f'} \hbar\_{\phi} \right) = \frac{1}{N+1} \sum\_{j=0}^{N} \left[ \sum\_{i=0}^{k} (f\_i)\_{\eta = j\pi'} \sum\_{i=0}^{k} (\theta\_i)\_{\eta = j\pi} \right]^2. \tag{27}
$$

The optimal values of the convergence control parameters are *hf* = −0.32677, *hθ* = −0.56129 and *hφ* = −0.46129, when *α*1 = *α*2 = *β* = 0.1, *Re* = *γ* = *S* = 0.2, *A* = 1.5, *Rd* = 0.4, *Q* = 0.2, *Sc* = 1.2, *St* = 0.3, Pr = 1. The values of convergence control parameters are choosen very carefully. The admissible ranges of parameters are taken. The results are convergen<sup>t</sup> within the ranges of these values. The values assigned to the fluid parameters are chosen carefully to satisfy the convergence criteria of OHAM. Beyond these values, the solution might not converge. One can see the total residual error in Figure 1.

approximations.
